A maximal inequality for continuous martingales and $M$-estimation in a Gaussian white noise model

Abstract

Some sufficient conditions to establish the rate of convergence of certain $M$-estimators in a Gaussian white noise model are presented. They are applied to some concrete problems, including jump point estimation and nonparametric maximum likelihood estimation, for the regression function. The results are shown by means of a maximal inequality for continuous martingales and some techniques developed recently in the context of empirical processes.